Discrete distributions in the extended FGM family: The p.g.f. approach

In this article the probability generating functions of the extended Farlie–Gumbel–Morgenstern family for discrete distributions are derived. Using the probability generating function approach various properties are examined, the expressions for probabilities, moments, and the form of the conditional distributions are obtained. Bivariate version of the geometric and Poisson distributions are used as illustrative examples. Their covariance structure and estimation of parameters for a data set are briefly discussed. A new copula is also introduced.     

CONDITIONAL EXPECTATION FORMULAE FOR COPULAS

Not only are copula functions joint distribution functions in their own right, they also provide a link between multivariate distributions and their lower‐dimensional marginal distributions. Copulas have a structure that allows us to characterize all possible multivariate distributions, and therefore they have the potential to be a very useful statistical tool. Although copulas can be traced back to 1959, there is still much scope for new results, as most of the early work was theoretical rather than practical. We focus on simple practical tools based on conditional expectation, because such tools are not widely available. When dealing with data sets in which the dependence throughout the sample is variable, we suggest that copula‐based regression curves may be more accurate predictors of specific outcomes than linear models. We derive simple conditional expectation formulae in terms of copulas and apply them to a combination of simulated and real data.     

Do stock returns have an Archimedean copula?

The flexible class of Archimedean copulas plays an important role in multivariate statistics. While there is a large number of goodness-of-fit tests for copulas and parametric families of copulas, the question if a given data set belongs to an arbitrary Archimedean copula or not has not yet received much attention in the literature. This paper suggests a new, straightforward method to test whether a copula is an Archimedean copula without the need to specify its parametric family. We conduct Monte Carlo simulations to assess the power of the test. The approach is applied to (bivariate) joint distributions of stock asset returns. We find that, in general, stock returns may have Archimedean copulas.     

Limiting Tail Dependence Copulas

The limiting lower-tail dependence copula (LLTDC) is defined as the copula of random variables which are right-truncated at thresholds tending to their left endpoints. This article shows LLTDCs are truncation-invariant and belong to the Ahmadi-Clayton family. Accordingly, it follows that limiting upper-tail dependence copulas are members of the survival Ahmadi-Clayton family.     

Copulas checker-type approximations: Application to quantiles estimation of sums of dependent random variables

Abstract

Several approximations of copulas have been proposed in the literature. By using empirical versions of checker-type copulas approximations, we propose non parametric estimators of the copula. Under some conditions, the proposed estimators are copulas and their main advantage is that they can be sampled from easily. One possible application is the estimation of quantiles of sums of dependent random variables from a small sample of the multivariate law and a full knowledge of the marginal laws. We show that estimations may be improved by including in an easy way in the approximated copula some additional information on the law of a sub-vector for example. Our approach is illustrated by numerical examples.     

Copules archimédiennes et families de lois bidimensionnelles dont les marges sont données

Every bivariate distribution function with continuous marginals can be represented in terms of a unique copula, that is, in terms of a distribution function on the unit square with uniform marginals. This paper is concerned with a special class of copulas called Archimedean, which includes the uniform representation of many standard bivariate distributions. Conditions are given under which these copulas are stochastically ordered and pointwise limits of sequences of Archimedean copulas are examined. We also provide two new one-parameter families of bivariate distributions which include as limiting cases the Frechet bounds and the independence distribution.     

Sampling nested Archimedean copulas

We give algorithms for sampling from non-exchangeable Archimedean copulas created by the nesting of Archimedean copula generators, where in the most general algorithm the generators may be nested to an arbitrary depth. These algorithms are based on mixture representations of these copulas using Laplace transforms. While in principle the approach applies to all nested Archimedean copulas, in practice the approach is restricted to certain cases where we are able to sample distributions with given Laplace transforms. Precise instructions are given for the case when all generators are taken from the Gumbel parametric family or the Clayton family; the Gumbel case in particular proves very easy to simulate.     

Gluing Copulas

We present a new way of constructing n-copulas, by scaling and gluing finitely many n-copulas. Gluing for bivariate copulas produces a copula that coincides with the independence copula on some grid of horizontal and vertical sections. Examples illustrate how gluing can be applied to build complicated copulas from simple ones. Finally, we investigate the analytical as well as statistical properties of the copulas obtained by gluing, in particular, the behavior of Spearman's ρ and Kendall's τ.     

Rectangular Patchwork for Bivariate Copulas and Tail Dependence

We present a method for constructing bivariate copulas by changing the values that a given copula assumes on some subrectangles of the unit square. Some applications of this method are discussed, especially in relation to the construction of copulas with different tail dependencies.     

On the identifiability of copulas in bivariate competing risks models

In competing risks models, the joint distribution of the event times is not identifiable even when the margins are fully known, which has been referred to as the “identifiability crisis in competing risks analysis” (Crowder, 1991). We model the dependence between the event times by an unknown copula and show that identification is actually possible within many frequently used families of copulas. The result is then extended to the case where one margin is unknown. The Canadian Journal of Statistics 41: 291–303; 2013 © 2013 Statistical Society of Canada     

Risk analysis in the brazilian stock market: copula-APARCH modeling for value-at-risk

Risk management of stock portfolios is a fundamental problem for the financial analysis since it indicates the potential losses of an investment at any given time. The objective of this study is to use bivariate static conditional copulas to quantify the dependence structure and to estimate the risk measure Value-at-Risk (VaR). There were selected stocks that have been performing outstandingly on the Brazilian Stock Exchange to compose pairs trading portfolios (B3, Gerdau, Magazine Luiza, and Petrobras). Due to the flexibility that this methodology offers in the construction of multivariate distributions and risk aggregation in finance, we used the copula-APARCH approach with the Normal, T-student, and Joe-Clayton copula functions. In most scenarios, the results showed a pattern of dependence at the extremes. Moreover, the copula form seems not to be relevant for VaR estimation, since in most portfolios the appropriate copulas lead to significant VaR estimates. It has found that the best models fitted provided conservative risk measures, estimates at 5% and 1%, in a scenario more aggressive.     

Copulas with Truncation-Invariance Property

For a truncation-invariant copula, truncation does not change the dependence structure as well as all nonparametric measures of association such as Kendall's tau and Spearman's rho. In this article, we show that the products of algebraically independent Archimedean multivariate Clayton copulas and standard uniform distributions are the only truncation-invariant copulas.     

Copulas related to piecewise monotone functions of the interval and associated processes

In this work, we derive the copulas related to vectors obtained from the so-called chaotic stochastic processes. These are defined by the iteration of certain piecewise monotone functions of the interval [0, 1] to some initial random variable. We study some of its properties and present some examples. Since often these types of copulas do not have closed formulas, we provide a general approximation method which converges uniformly to the true copula. Our results cover a wide class of processes, including the so-called Manneville–Pomeau processes. The general theory is applied to the parametric estimation in certain chaotic processes. A Monte Carlo simulation study is also presented.     

A stochastic representation and sampling algorithm for nested Archimedean copulas

A general sampling algorithm for nested Archimedean copulas was recently suggested. It is given in two different forms, a recursive or an explicit one. The explicit form allows for a simpler version of the algorithm which is numerically more stable and faster since less function evaluations are required. The algorithm can also be given in general form, not being restricted to a particular nesting such as fully nested Archimedean copulas. Further, several examples are given.     

A Decomposition of Copulas and Its Use

ABSTRACT

In this article, we create a decomposition that represents and describes the depen-dence structure between two variables. Since copulas provide a deep understanding of the dependence structure by eliminating the effects of the marginals, they play a key role in this study. We define a discretized copula density matrix and decompose it into a set of permutation matrices by using the Birkhoff–von Neumann theorem. This decomposition provides a way to effectively apply the concepts of copulas to solve problems in multivariate statistical data analysis.     

Measures of tail asymmetry for bivariate copulas

Three tail asymmetry measures for bivariate copulas are introduced using two different approaches—univariate skewness of a projection and distance between a copula and its survival/reflected copula. We compare the asymmetry measures based on certain desirable properties. Bounds for each measure are obtained and also copulas which attain these extreme values are identified. Two data examples show the amount of asymmetry that might be expected in practice.     

Predictive assessment of copula models

Copulas are powerful explanatory tools for studying dependence patterns in multivariate data. While the primary use of copula models is in multivariate dependence modelling, they also offer predictive value for regression analysis. This article investigates the utility of copula models for model‐based predictions from two angles. We assess whether, where, and by how much various copula models differ in their predictions of a conditional mean and conditional quantiles. From a model selection perspective, we then evaluate the predictive discrepancy between copula models using in‐sample and out‐of‐sample predictions both in bivariate and higher‐dimensional settings. Our findings suggest that some copula models are more difficult to distinguish in terms of their overall predictive power than others, and depending on the quantity of interest, the differences in predictions can be detected only in some targeted regions. The situations where copula‐based regression approaches would be advantageous over traditional ones are discussed using simulated and real data. The Canadian Journal of Statistics 47: 8–26; 2019 © 2018 Statistical Society of Canada     

Applications and asymptotic power of marginal-free tests of stochastic vectorial independence

Fully nonparametric tests for the independence between random vectors are studied in this paper. The test statistics are functionals of an empirical process defined as the difference between the joint empirical copula and the product of the empirical copulas associated to the vectors that are suspected to be independent. The validity of a weighted bootstrap procedure is established, which allows for a quick computation of p-values. A special attention is given to the asymptotic behavior of the tests under contiguous sequences of distributions. Finally, a characteristic of the copulas in the Archimedean class in terms of independence of vectors is exploited in order to propose a new goodness-of-fit procedure.     

The Joy of Copulas: Bivariate Distributions with Uniform Marginals

We describe a class of bivariate distributions whose marginals are uniform on the unit interval. Such distributions are often called “copulas.” The particular copulas we present are especially well suited for use in undergraduate mathematical statistics courses, as many of their basic properties can be derived using elementary calculus. In particular, we show how these copulas can be used to illustrate the existence of distributions with singular components and to give a geometric interpretation to Kendall's tau.     

Estimating non-simplified vine copulas using penalized splines

Vine copulas (or pair-copula constructions) have become an important tool for high-dimensional dependence modeling. Typically, so-called simplified vine copula models are estimated where bivariate conditional copulas are approximated by bivariate unconditional copulas. We present the first nonparametric estimator of a non-simplified vine copula that allows for varying conditional copulas using penalized hierarchical B-splines. Throughout the vine copula, we test for the simplifying assumption in each edge, establishing a data-driven non-simplified vine copula estimator. To overcome the curse of dimensionality, we approximate conditional copulas with more than one conditioning argument by a conditional copula with the first principal component as conditioning argument. An extensive simulation study is conducted, showing a substantial improvement in the out-of-sample Kullback–Leibler divergence if the null hypothesis of a simplified vine copula can be rejected. We apply our method to the famous uranium data and present a classification of an eye state data set, demonstrating the potential benefit that can be achieved when conditional copulas are modeled.